To find $g(x)$, which is the translation of $f(x) = x^2$ by 5 units to the right and 6 units down, we can apply the rules for translating a function:

1. Translation 5 units to the right: To translate a function $f(x)$ to the right by $h$ units, we replace $x$ with $x - h$. For 5 units to the right, we replace $x$ with $x - 5$.

   $f(x) = x^2 \quad \Rightarrow \quad f(x - 5) = (x - 5)^2$

2. Translation 6 units down: To translate the function 6 units down, we subtract 6 from the entire function.

   $g(x) = (x - 5)^2 - 6$

So, the translated function is:

$g(x) = (x - 5)^2 - 6$

Would you like any further details on how these transformations work?

Here are some related questions you might find interesting:

1. How would the translation of $f(x) = x^2$ be affected if it were 5 units left and 6 units up instead?
2. What would happen if we applied a vertical stretch by a factor of 2 to $g(x)$?
3. How do horizontal and vertical translations differ in their effect on the graph of a function?
4. What is the vertex of the parabola described by $g(x) = (x - 5)^2 - 6$?
5. How can you find the domain and range of the function $g(x)$?

Tip: When transforming functions, remember that horizontal translations involve replacing $x$ with $x - h$ (where $h$ is the shift), while vertical translations involve adding or subtracting a constant outside the function.